Victor S. L vov, Baku, January 21, Turbulent Drag Reduction by Dilute Solution of Polymers

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1 Victor S. L vov, Baku, January 21, 2009 Turbulent Drag Reduction by Dilute Solution of Polymers In collaboration with T.S. Lo (TL), A. Pomyalov (AP), I. Procaccia (IP), V. Tiberkevich (VT). Weizmann Inst., Rehovot, Israel Roberto Benzi, Elizabeth de Angelis, Carlo Casciola Rome Univ., Italy and Emily Ching Hong Kong Univ. 1

2 Review of our theory: IP, VL, and R. Benzi, Theory of drag reduction by polymers in wall bounded turbulence,, Reviews of Modern Physics, 80, (2008), Details in: 1. VL, AP, IP & VT, Drag Reduction by Polymers in Wall Bounded Turbulence, Phys. Rev. Lett, 92, ( 2004). 2. E. de Angelis, C. Casciola, VL, AP, & VT, Drag Reduction by a linear viscosity profile, Phys. Rev. E, 70, (R) (2004). 3.R. Benzi, VL, IP & VT, Saturation of Turbulent Drag Reduction in Dilute Polymer Solutions, EuroPhys. Lett, (2004). 4.VL, AP, IP & VT, The polymer stress tensor in turbulent shear flow, Phys. Rev. E., 71, (2005). 5.VL, AP & VT, Simple analytical model for entire turbulent boundary layer over flat plane, Environmental Fluid Mechanics, 5, (2005). 6.R. Benzi, E. de Angelis, VL, IP & VT. Maximum Drag Reduction Asymptotes and the Cross-Over to the Newtonian plug, Journ. of Fluid Mech., 551, 185 (2006) 7.R. Benzi, E.S.C. Ching, TL, VL & IP. Additive Equivalence in Turbulent Drag Reduction by Flexible & Rodlike Polymers Phys. Rev. E 72, (2005). 8.R. Benzi, E. de Angelis, VL & IP. Identification and Calculation of the Universal Maximum Drag Reduction Asymptote by Polymers in Wall Bounded Turbulence, Phys. Rev. Letts., 95 No 14, (2005) 9. Y. Amarouch ene, D. Bonn, H. Kellay, TL, VL, IP, Reynolds number dependence of drag reduction by rod-like polymers, J. Fluid Mech., submitted, Also nlin.cd/

3 ... the increase in water flow by adding only 30 wppm of Polyethylene oxide (PEO ) allows firemen to deliver the same amount of water to a higher elevation from a 1.5-inch hose which can be handled by two men, instead of four men to handle 2.5-inch hose fed only by water... 3

4 History: experiments, engineering developments, ideas & problems B.A.Toms, 1949: An addition of 10 4 weight parts of long-chain polymers can suppress the turbulent friction drag up to 80%. This phenomenon of drag reduction is intensively studied (by 1995 there were about 2500 papers, now we have many more) and reviewed by Lumley (1969), Hoyt (1972), Landhal (1973), Virk (1975), McComb (1990), de Gennes (1990), Sreenivasan & White (2000), and others. In spite of the extensive and continuing activity the fundamental mechanism has remained under debate for a long time, oscillating between Lumley s suggestion of importance of the polymeric contribution to the fluid viscosity and de Gennes s idea of importance of the polymeric elasticity. Some researches tried to satisfy simultaneously both respectable. Nevertheless, the phenomenon of drag reduction has various technological applications from fire engines (allowing a water jet to reach high floors) to oil pipelines, starting from its first and impressive application in the Trans-Alaska Pipeline System. 4

5 Trans-Alaska Pipeline System L 800 Miles, = 48 inches. TAPS was designed with 12 pump stations (PS) and a throughput capacity of 2.00 million barrels per day (BPD).. Now TAPS operates with only 10 PS (final 2 were never build) with throughput 2.1 BPD, with a total injection of 250 wppm of polymer PEO drag reduction additive (wppm weight parts per million,. 250 wppm = ) 5

6 INFO PEO page: Typical parameters of polymeric molecules PEO Polyethylene oxide (N [ CH 2 CH 2 O] ) and their solutions in water: degree of polymerization N (1.2 12) 10 4, molecular weight M (0.5 8) 10 6, equilibrium end-to-end distance R 0 (7 20) 10 8 m, maximal end-to-end distance R max NR m; typical mass loading ψ = For ψ = of PEO ν pol ν 0. PEO solutions is dilute up to ψ =

7 Essentials of the phenomenon: Virk s universal MDR asymptote.. & cross-over to Newtonian plug V + 50 Newtonian, Warholic, 1999 MDR, Rollin, 1972 Rudd, Rollin, 1972 DNS, De Angelis, 2003 Newtonian plugs y +. Mean normalized velocity profiles as a function of the normalized dis tance from the wall in drag reduction The green circles DNS fo Newtonian channel flow, open circle experiment. The Prandtl-Karman log-profile : V + = 2.5ln y The red squares experiment, univer sal Virk s MDR asymptote V + = 11.7ln y The blue triangles & green open triangles -over, for intermediate concentrations of the polymer, from the MDR asymptote to the Newtonian plug Wall normalization: Re L p L ν 0, y + yre L, V + V p L. 7

8 Asymptotical Universality of Drag Reduction by Polymers in Wall Bounded Turbulence: Outline: History of the problem Essentials of the phenomenon subject of the theory We are here A theory of drug reduction and its verification:. Simple theory of basic phenomena in drag reduction. Origin & calculation of Maximum Drag Reduction Asymptote. DNS verification of the the simple theory of drag reduction Advanced approach. Elastic stress tensor & effective viscosity. -over from the MDR asymptote to the Newtonian plug Summary of the theory 8

9 Simple theory of basic phenomena in drag reduction is based on: An approximation of effective polymeric viscosity for visco-elastic flows, An algebraic Reynolds-stress model for visco-elastic wall turbulence. An approximation of effective polymeric viscosity accounts for the effective polymeric viscosity (according to Lumley) and neglects elasticity (accounting for the elasticity effects was the main point of the de Gennes approach) We stress: the polymeric viscosity is r-dependent, Lumley s ν p ν p (r) Algebraic Reynolds-stress model for a channel (of width 2L): Exact (standard) equation for the flux of mechanical momentum: ν(y)s(y) + W(y) = p L, ν(y) ν 0 + ν p (y). (1) Hereafter: x & y streamwise & wall-normal directions, p dp/dx mean shear: S(y) d V x(y), Reynolds stress: W(y) v x v y. dy 9

10 Simple theory of basic phenomena in drag reduction: An approximation of effective polymeric viscosity ν 0 ν(y) ν 0 +ν p (y) Algebraic Reynolds-stress model for viscoelastic flows: Eq. for the flux of mechanical momentum: (for a channel of width 2L) ν(y)s(y) + W(y) = p L, ν(y) ν 0 + ν p (y). (1) p dp/dx, S(y) d V x (y)/dy, W(y) v x v y. Balance Eq. for the density of the turbulent kinetic energy K v 2 /2: [ ν(y)(a/y) 2 + b Simple TBL closure: W(y) K(y) = K(y)/y ] c 2 N, K(y) = W(y)S(y), (2) for Newtonian flow, c 2 V, for viscoelastic flow. (3) Dynamics of polymers (in harmonic approximation, with τ p polymeric relaxation time) restricts the level of turbulent activity, (consuming kinetic energy) at the threshold level: 1 τ p ui u i W(y) τ p. (4) r j r j y 9-a

11 Algebraic Reynolds-stress model in wall units: Re L p L/ν 0, y + yre/l, V + V/ p L, ν + = [ 1 + +ν + p ]. Mechanical balance: ν + S + + W + = 1, (1) Energy balance: ν + ( δ/y +) 2 + W + /κ K y + = S +, (2) Polymer dynamics: W + = L 2 ν 0 y + 2 /τ p Re 2. (3) Test case: Newtonian turbulence: Disregard polymeric terms: ν + 1 & solve quadratic Eqs. (1)-(2) for S + (y + ) & integrate. The result: For y + δ : V + = y +. (4a) For y + δ : V + (y + ) = 1 ln Y (y + ) + B (y + ), B = 2δ 1 ln κ K κ K Y (y + ) = [y + + y +2 δ 2 + (2κ K ) 2 ]/2, [ e(1 + 2 κk δ) 4κ K ], (4b) (y + ) = 2 κ2 K δ2 + 4κ K [Y (y + ) y + ] + 1 2κ 2 K y+. Two fit parameters: κ N & δ. 10

12 V + Comparison of analytical profile (4) with experiment & numerics Experiment DNS Our model y +. using κ 1 K = 0.4 and δ N = 6 Summary: Our simple Algebraic Reynolds-stress model, based on the exact balance of mechanica momentum and K41 in spired model equation fo the local energy balance gives physically transparent analytical, semi-quantitative description of turbulent boundary layer. 11

13 Viscoelastic turbulent flow & Universal MDR asymptote: Mechanical balance ν + S + + W + = 1, (1) Energy balance: Polymer dynamics: ( ν + δn,v y + W + = L2 ν 0 τ p 2 ) 2 + y + W + κ K y + = S+, (2) Re 2 0 at fixed y+ & Re. (3) Equation (3) dictates: Maximum Drag Reduction (MDR) asymptote Re, W + = 0. We have learned that in the MDR regime: normalized (by wall units) turbulent kinetic energy [Eq.(3)]. K + W + 0; the mechanical balance [Eq.(1)] and the balance of kinetic energy. [Eq.(2)] are dominated by the polymeric contribution ν +. 12

14 Viscoelastic turbulent flow & Universal MDR asymptote: Mech. & energy bal.: ν + S + + W + = 1, ν + ( δn,v Polymer dynamics: W + = L2 ν 0 τ p 2 y + y + ) 2 + W + κ K y + = S+,(1,2) Re 2 0 at fixed y+ & Re. (3) Maximum Drag Reduction (MDR) asymptote Re, W + = 0. In the MDR regime Eqs. (1,2) become ν + S + = 1, ν + δ 2 V = S+ y +2 and have solution: S + = δ V /y +, ν + = y + /δ V for y + δ V, because ν + (y + ) ν + 0 = 1. y+ For y + δ V, S + = 1, ν + = 1. Integration V + (y + ) = δ V Universal MDR asymptote: V + (y + ) = δ V ln (e y + ) /δ V δ V S + (ξ)dξ. (4) 12-a

15 Viscoelastic turbulent flow & Universal MDR asymptote: Mech. & energy bal.: ν + S + + W + = 1, ν + ( δn,v Polymer dynamics: W + = L2 ν 0 τ p 2 y + y + ) 2 + W + κ K y + = S+,(1,2) Re 2 0 at fixed y+ & Re. (3) At MDR: W + = 0 Eqs. (1,2) ν + S + = 1, ν + δ 2 V = S+ y +2 S + = δ V /y +, ν + = y + /δ V V + (y + ) = δ V ln ( e y + /δ V ). (4) Summary: In the MDR regime normalized (by wall units) turbulent kinetic energy. K + 0; MDR regime is the edge of turbulent solution of the Navier-Stokes. Eq. (NSE) with the largest possible effective viscosity ν(y),. at which the turbulence still exists! MDR profiles of ν(y) & S(y) are determined by the NSE itself and. are universal, independent of parameters of polymeric additives. 12-b

16 Calculation of δ V in the MDR asymptote: V + (y + ) = δ V ln ( e y + /δ V ). Consider ν + S + + W + = 1, y + +2 κ K y + = S+ with prescribed ν + = 1 + α(y + δ N ) and replace flow dependent δ N,V (α) with yet arbitral α: ν +δ2 N,V W + [1 + α(y + δ N )]S + +W + = 1, [1 + α(y + δ N )] 2 (α) y +2 + W + κ K y + = S+ ( ). Clearly, δ N = (0) (Newtonian flow) and δ V = (α V ), where (α V ) is the MDR solution of (*) in asymptotical region y + 1 with W = 0: α V (α V ) = 1, (α) = δ N 1 αδ N, α V = 1 2δ N, δ V = 2 δ N. (α) follows from the requirement of the rescaling symmetry of Eq. (*): y + y y+ g( δ), g( δ) 1 + α( δ δ N ), δ δ δ g( δ), S+ S S + g( δ). Finally: V + (y + ) = 2δ N ln e y+ 2 δ N, with Newtonian constant δ N 6 ( ) 13

17 Virk s MDR asymptote: experiment ( ) vs our equation ( ) V Newtonian, Warholic, 1999 MDR, Rollin, 1972 Rudd, Rollin, 1972 DNS, De Angelis, 2003 MDR, our theory 30 Newtonian, our theory Newtonian plugs The red squares experiment MDR: V + = 11.7ln y ( V + = 2δ N ln ( e y + /2δ N ). ( Taking δ N 6 from Newtonia data one has slope 2δ N 1 close to 11.7 in ( ) and intercep. 2δ N ln(e/2δ N ) 17.8, close in ( ). y +. Summary: Maximum possible drag reduction (MDR asymptote) corresponds to the maximum possible viscosity profile at the edge of existence of turbulent solution and thus is universal, i.e. independent of polymer parameters, if polymers are able to provide required viscosity profile. 14

18 Renormalized NSE DNS test of the scalar mean viscosity model Recall: in the MDR regime: ν p (y) y ν/ν V x y/l y + The viscosity (Left) & NSE-DNS mean velocity profiles (Right). Re = 6000(with centerline velocity). Solid black line standard Newtonian flow. One sees a drag reduction in the scalar mean viscosity model. 15

19 Comparison of renormalized NSE model ( Red line : ) and full FENE-P model ( Blue circles: o o o o ) [Newtonian flow: -] Re = 6000 for Mean velocity profiles (Left) & Stream-wise turbulent velocity (Right) 20 3 V V x y y/l Conclusion: Suggested simple model of polymer suspension with selfconsistent viscosity profile really demonstrates the drag reduction itself and its essentials: mean velocity, kinetic energy profiles not only in the MDR regime, but also for intermediate Re. 16

20 Riddle Intuitively: effective polymeric viscosity ν p (y) should be proportional to the (thermodynamical) mean square polymeric extension R R 2, averaged over turbulent assemble, R 0 = R : ν p (y) R 0 (y). However, in the MDR regime in our model ν p (y) increases with the distance from the wall, ν p (y) y, while experimentally R(y) decreases. A way out Instead of intuitive (and wrong) relationship ν p (y) R 0 (y) one needs to find correct connection between ν p (y) and mean polymeric conformation tensor R ij 0 R i R j. This is a goal of Advanced approach: Elastic stress tensor Π & effective viscosity 17

21 Advanced approach: Elastic stress tensor Π & effective viscosity Define: elastic stress Π ij R ij ν 0,p /τ p & conformation R ij R i R j tensors,. polymeric laminar viscosity ν 0,p & polymeric relaxation time τ p.. R = r/r 0 end-to-end distance, normalized by its equilibrium value Write: Navier Stokes Equation (NSE) for dilute polymeric solutions: v t + ( v ) v = ν 0 v P + Π, (1). together with the equation for the elastic stress tensor: Π t + (v )Π = SΠ + ΠS 1 τ p ( Π Πeq ), S ij v i / x j, (2) Averaging Eq. (1) and y 0... dỹ one has equation for S(y) v x / y : ν 0 S(y)+Π xy 0 (y)+w(y) = p L, (3) in which Π xy 0 (y) Πxy (y) is the momentum flux, carried by polymers. 18

22 Elastic stress tensor Π & effective viscosity in the MDR regime In short: Π ij R ij ν 0,p /τ p, polymeric viscosity ν 0,p & relaxation time τ p. NSE for dilute polymer solutions: v ( ) t + v v = ν 0 v P + Π, (1) Π t + (v )Π = SΠ + ΠS 1 ( ) Π Π τ eq, S ij v i / x j, (2) p Eq. (1) gives Eq. for the mean shear: ν 0 S(y) + Π xy 0 (y) + W(y) = p L, (3) Averaging Eq. (2), and taking D/Dt = 0 one gets stationary Eq. for Π 0 : Π 0 = τ p ( S0 Π 0 + Π 0 S 0 + Q), Q τ 1 p Π eq + s π + π s. (4a) In the shear geometry S 0 S 0 = 0. This helps to find by the subsequent substitution of the RHS RHS of Eq. (4a) its solution: Π 0 = 2 τ 3 p S 0 Q S 0 +τ2 p( S0 Q+Q S 0) +τp Q. (4b) 18-a

23 Elastic stress tensor Π & effective viscosity in the MDR regime In short: Π ij R ij ν 0,p /τ p, polymeric viscosity ν 0,p & relaxation time τ p. NSE for dilute polymer solutions: v ( ) t + v v = ν 0 v P + Π, (1) Π t + (v )Π = SΠ + ΠS 1 ( ) Π Π τ eq, S ij v i / x j, (2) p Eq. (1) gives Eq. for the mean shear: ν 0 S(y) + Π xy 0 (y) + W(y) = p L, (3) ( ) Eq. (2) gives Eq. for Π 0 : Π 0 = 2 τ p 3 S 0 Q S 0 + τ2 S p 0 Q + Q S 0 + τ p Q. (4b) At the onset of drag reduction Deborah number at the wall De 0 1, De 0 = De(0), De(y) τ p S(y). In the MDR regime: De(y) 1. In the limit De(y) 1, Eq. (4b) gives: Π 0 (y) = Π yy 0 (y) 2[De(y)] 2 De(y) 0 De(y) C, C = Q zz Q yy 1. For De(y) 1 tensorial structure of Π 0 becomes universal. In particular: Π xy 0 (4c) (y) = De(y) Πyy 0 (y). (4d) 18-b

24 Elastic stress tensor Π & effective viscosity in the MDR regime v ( ) NSE for dilute polymer solutions: t + v v = ν 0 v P + Π, (1) Π t + (v )Π = SΠ + ΠS 1 ( ) Π Π τ eq, S ij v i / x j, (2) p Eq. (1) gives Eq. for the mean shear: In the limit De(y) 1, Eq. (4b) gives: Π 0 (y) = Π yy 0 (y) ν 0 S(y)+Π xy 0 (y)+w(y) = p L, (3) 2[De(y)]2 De(y) 0 De(y) C, C 1. (4c) Π xy 0 (y) = De(y)Πyy 0 (y) = S(y)τ p Π yy (y). (4d) 0 Insertion (4d) into (3) the model eq: [ ν 0 +ν p (y)] S(y) + W(y) = p L, in which ν p (y) τ p Π yy 0 (y) = ν 0,p Ryy is the polymeric viscosity. (5) Summary: Effective polymeric viscosity ν p (y) is proportional to yy component of the conformation (and elastic stress) tensor, not to its trace. 18-c

25 Comparison of the theory with Direct Numerical Simulation R yy x 10 R ij R xx y/l We found: In the MDR regime, ν p (y) R yy (y) while S(y) 1/y, R yy (y) y R xx (y) = 2S 2 (y)τp 2 R yy (y) 1 y, DNS data for R yy red circles y +.. for R xx black squares are fitted by red y and black 1/y lines. Summary: Predicted spacial profiles of effective viscosity and polymeric extension are consistent with the DNS and experimental observations 19

26 Cross-over from the MDR asymptote to the Newtonian plug Model Eqs. [ ν0 + ν p (y)] S(y) + W(y) = p L, (1) {[ ν 0 + ν p (y)](a/y) 2 ]} + b K(y)/y K(y) = W(y)S(y). (2) W(y)/K(y) = c 2 V, τ2 p W(y) y2. (3,4) Reminder: In the MDR regime red-marked terms in Eqs. (1,2) are small. -over of linearly extended polymers: Eq. (1) p L W(y ) with Eq. (4): p L y 2 /τ2 p y τ p p L y + De(0). (5a) ( ) -over of finite extendable polymers: ν p (y) ν p,max ν 0 c p a 3 Np. In the MDR: ν p (y + ) ν 0 y + y + c ( ) p a 3 Np. (5b) ( a Np ) 3 In general: y + De(0) c p ( ) De(0) + c p a 3. (6) Np Verification: -over (5a) is consistent with DNS of Yu et. al. (2001), -over (5b) is in agreement with DNA experiment of Choi et. al. (2002) 20

27 SUMMARY OF THE RESULTS Essentials of the drag reduction by dilute elastic polymers can be understood within the suggested Effective Viscosity. Approximation. For the NSE with the effective viscosity we have suggested an Algebraic Reynolds-stress model that describes relevant characteristics of the Newtonian and viscoelastic turbulent flows in agreement with available DNS and experimental data. The model allows one to clarify the origin of the universality of the maximum possible drag reduction and to calculate universal Virk s constants, that are are a good quantitative agreement with the experiments. The model predicts two mechanisms of -over MDR Newtonian plug in a qualitative agreement with DNS and experiment. In short: Basic physics of drag reduction in polymeric solutions is understood and has reasonable simple and transparent description in the framework of developed theory. Further developments and detailing are possible T HE EN D 21-d

Maximum drag reduction asymptotes and the cross-over to the Newtonian plug

J. Fluid Mech. (26), vol. 551, pp. 185 195. c 26 Cambridge University Press doi:1.117/s221125795 Printed in the United Kingdom 185 Maximum drag reduction asymptotes and the cross-over to the Newtonian